Turbine nozzle airfoil profile

ABSTRACT

The present application provides a turbine nozzle including an airfoil shape. The airfoil shape may have a nominal profile substantially in accordance with Cartesian coordinate values of X, Y and Z set forth in Table I. The Cartesian coordinate values of X, Y and Z are non-dimensional values from 0% to 100% convertible to dimensional distances in inches by multiplying the Cartesian coordinate values of X, Y and Z by a height of the airfoil in inches. The X and Y values, when connected by smooth continuing arcs, define airfoil profile sections at each distance Z. The airfoil profile sections at Z distances may be joined smoothly with one another to form a complete airfoil shape.

FIELD

The present application relates generally to a turbine nozzle for a gas turbine engine, and more particularly relates to a turbine nozzle airfoil profile for a turbine nozzle.

BACKGROUND

Some aircraft and/or power plant systems, for example certain jet aircraft, nuclear, simple cycle and combined cycle power plant systems, employ turbines (also referred to as turbomachines) in their design and operation. Some of these turbines employ airfoils (e.g., stator vanes or nozzles and rotor blades.) which during operation are exposed to fluid flows. These airfoils are configured to aerodynamically interact with the fluid flows and generate energy (e.g., creating thrust, turning kinetic energy to mechanical energy, thermal energy to mechanical energy, etc.) from these fluid flows as part of power generation. As a result of this interaction and conversion, the aerodynamic characteristics and losses of these airfoils have an impact on system and turbine operation, performance, thrust, efficiency, and power.

BRIEF DESCRIPTION

Aspects and advantages are set forth below in the following description, or may be obvious from the description, or may be learned through practice.

The present application provides a turbine nozzle including an airfoil shape. The airfoil shape may have a nominal profile substantially in accordance with Cartesian coordinate values of X, Y and Z set forth in Table I. The Cartesian coordinate values of X, Y and Z are non-dimensional values from 0% to 100% convertible to dimensional distances in inches by multiplying the Cartesian coordinate values of X, Y and Z by a height of the airfoil in inches. The X and Y values, when connected by smooth continuing arcs, define airfoil profile sections at each distance Z. The airfoil profile sections at Z distances being joined smoothly with one another to form a complete airfoil shape.

The present application further provides a turbine nozzle including an airfoil having a suction-side uncoated nominal airfoil profile substantially in accordance with suction-side Cartesian coordinate values of X, Y and Z set forth in Table I. The Cartesian coordinate values of X, Y and Z are non-dimensional values from 0% to 100% convertible to dimensional distances in inches by multiplying the Cartesian coordinate values of X, Y and Z by a height of the airfoil in inches. The X and Y values, when connected by smooth continuing arcs, define airfoil profile sections at each Z distance. The airfoil profile sections at the Z distances may be joined smoothly with one another to form a complete suction-side airfoil shape. The X, Y and Z distances being scalable as a function of the same constant or number to provide a scaled-up or scaled-down airfoil.

The present application further provides a turbine with a number of turbine nozzles having an airfoil having an airfoil shape. The airfoils having a nominal profile substantially in accordance with Cartesian coordinate values of X, Y and Z set forth in Table I. The Cartesian coordinate values of X, Y and Z are non-dimensional values from 0% to 100% convertible to dimensional distances in inches by multiplying the Cartesian coordinate values of X, Y and Z by a height of the airfoil in inches. The X and Y values, when connected by smooth continuing arcs, define airfoil profile sections at each Z distance. The airfoil profile sections at the Z distances may be joined smoothly with one another to form a complete airfoil shape.

Those of ordinary skill in the art will better appreciate the features and aspects of such embodiments, and others, upon review of the specification.

BRIEF DESCRIPTION OF THE DRAWINGS

A full and enabling disclosure of various embodiments, including the best mode thereof to one skilled in the art, is set forth more particularly in the remainder of the specification, including reference to the accompanying figures, in which:

FIG. 1 is a schematic representation of an exemplary turbine having multiple stages with each stage including alternating rows of turbine rotor blades and turbine nozzles according to at least one embodiment of the present disclosure;

FIG. 2 is a perspective view of an exemplary turbine nozzle according to at least one embodiment of the present disclosure; and

FIG. 3 provides a cross-sectional view of an airfoil portion of the turbine nozzle of FIG. 2.

DETAILED DESCRIPTION

Referring now to the drawings, particularly to FIG. 1, there is illustrated an exemplary turbine 10 of a gas turbine 12 including a plurality of turbine stages arranged in serial flow order. Each stage of the turbine includes a row of turbine nozzles or vanes disposed axially adjacent to a corresponding row of turbine rotor blades. Four stages are illustrated in FIG. 1. The exact number of stages of the turbine 10 is a choice of engineering design, and may be more or less than the four stages illustrated in FIG. 1. The four stages are merely exemplary of one turbine design, and are not intended to limit the presently claimed turbine rotor blade in any manner.

The first stage comprises a plurality of turbine nozzles or vanes 100 and a plurality of turbine rotor blades 150. The turbine nozzles 100 are annularly arranged about an axis of a turbine rotor 14. The turbine rotor blades 150 are annularly arranged about and coupled to the turbine rotor 14. A second stage of the turbine 12 includes a plurality of turbine nozzles 200 annularly arranged about the axis of the turbine rotor 14 and a plurality of turbine rotor blades 250 annularly arranged about and coupled to the turbine rotor 14. The turbine nozzles 200 may be coupled to a casing 16 of the turbine 12.

As shown in FIG. 1, a third stage of the turbine 12 includes a plurality of turbine nozzles 300 annularly arranged about the axis of the turbine rotor 14 and a plurality of turbine rotor blades 350 annularly arranged about and coupled to the turbine rotor 14. The turbine nozzles 300 may be coupled to the casing 16 of the turbine 12. In particular embodiments, the turbine 12 includes a fourth stage which includes plurality of turbine nozzles 400 annularly arranged about the axis of the turbine rotor 14 and a plurality of turbine rotor blades 450 annularly arranged about and coupled to the turbine rotor 14. The fourth stage turbine nozzles 400 may be coupled to the casing 16 of the turbine 12.

It will be appreciated that the turbine nozzles 100, 200, 300 and 400 and turbine rotor blades 150, 250, 350 and 450 are disposed or at least partially disposed within a hot gas path 18 of the turbine 12. The various stages of the turbine 10 at least partially define the hot gas path 18 through which combustion gases, as indicated by arrows 20, flow during operation of the gas turbine 12.

FIG. 2 provides a perspective view of an exemplary turbine nozzle 50 as may be incorporated in any stage of the turbine 12. FIG. 3 provides a cross-sectional top view of an airfoil 52 of the turbine nozzle 50 shown in FIG. 2. In one embodiment the turbine nozzle 50 as provided in Table I below is representative of turbine nozzle 300 of the plurality of turbine nozzles 300 of the third stage as shown in FIG. 1. As shown in FIGS. 2 and 3 collectively, the turbine nozzle 50 airfoil 52 includes a pressure-side wall 54 and an opposing suction-side wall 56. The pressure-side wall 54 and the suction-side wall 56 meet or intersect at a leading edge 58 and at a trailing edge 60 of the airfoil 52. A chord line 62 extends between the leading edge 58 and the trailing edge 60 such that pressure and suction-side walls 54, 56 can be said to extend in chord or chordwise between the leading edge 58 and the trailing edge 60.

As shown in FIG. 2, the airfoil 52 includes a root or first end 64 which intersects with and extends radially outwardly from an inner band 66 of the turbine nozzle 50. The airfoil 52 terminates radially at a second end or tip 68 of the airfoil 52. The second end 68 intersects with and extends radially inwardly from an outer band 70 of the turbine nozzle 50. The pressure and suction-side walls 54, 56 can be said to extend in height and/or span or in a span-wise direction 72 between the root 64 and/or the inner band 66 and the second end 68 or the outer band 70. In other words, each turbine nozzle 50 includes an airfoil 52 having opposing pressure and suction-side walls 54, 56 that extend in chord or chordwise 62 between opposing leading and trailing edges 58, 60 and that extend in span or span-wise 72 between the root 64 and the second end 68 of the airfoil 52.

In particular configurations, the airfoil 52 may include a fillet 74 formed between the inner band 66 and the airfoil 52 proximate to the root 64. The fillet 74 can include a weld or braze fillet, which can be formed via conventional MIG welding, TIG welding, brazing, etc., and can include a profile that can reduce fluid dynamic losses as a result of the presence of fillet 74. In particular embodiments, the inner band 66, airfoil 52 and the fillet 74 can be formed as a single component, such as by casting and/or machining and/or 3D printing and/or any other suitable technique now known or later developed and/or discovered.

The airfoil 52 of the turbine nozzle 50 has a profile at any cross-section taken between the inner band 66 or the root 64 and the second end 68. The X, Y, and Z values are given in Table I as percentage values of the airfoil length or height/span. As one example only, the height or radial span of the airfoil 52 of turbine nozzle 50 may be from about 5 inches to about 20 inches (about 12 centimeters to about 51 centimeters). However, it is to be understood that heights below or above this range may also be employed as desired in the specific application.

The gas turbine hot gas path 18 requires airfoils 52 that meet system requirements of aerodynamic and mechanical blade loading and efficiency. To define the airfoil shape of each turbine nozzle airfoil 52, there is a unique set or loci of points in space that meet the stage requirements and can be manufactured. These unique loci of points meet the requirements for stage efficiency and are arrived at by iteration between aerodynamic and mechanical loadings enabling the turbine to run in an efficient, safe and smooth manner. These points are unique and specific to the system.

The loci that defines the turbine nozzle airfoil profile includes a set of about 2,200 points with X, Y and Z dimensions relative to a reference origin coordinate system. The Cartesian coordinate system of X, Y and Z values given in Table I below defines the profile of the turbine nozzle airfoil at various locations along its length. Table I lists data for a non-coated airfoil. The envelope/tolerance for the coordinates is about +/−5% in a direction normal to any airfoil surface location and/or about +/−5% of the chord 62 in a direction nominal to any airfoil surface location. The point data origin 76 is defined at or proximate to the leading edge 58 of the airfoil 52 and at or proximate to the root 64.

The coordinate values for the X, Y and Z coordinates are set forth in non-dimensionalized units by the values provided in Table I although other units of dimensions may be used when the values are appropriately converted. The X, Y and Z values set forth in Table I are also expressed in non-dimensional form (X, Y, and Z) from 0% to 100% of the height or span of the airfoil 52. As one example only, the Cartesian coordinate values of X, Y and Z may be convertible to dimensional distances by multiplying the X, Y and Z values by a height of the airfoil at the leading edge 58 and multiplying by a constant number (e.g., 100). As one example only, the Cartesian coordinate values of X, Y and Z may be convertible to dimensional distances by multiplying the X, Y and Z values by a height of the airfoil at the trailing edge 60 and multiplying by a constant number (e.g., 100).

To convert the Z value to a Z coordinate value, e.g., in inches, the non-dimensional Z value given in Table I is multiplied by the Z length of the airfoil in inches. As described above, the Cartesian coordinate system has orthogonally-related X, Y and Z axes and the X axis lies generally parallel to the turbine rotor centerline, i.e., the rotary axis and a positive X coordinate value is axial toward an exhaust end of the turbine 14. The positive Y coordinate value extends tangentially in the direction of rotation of the rotor shaft 14 and the positive Z coordinate value is radially outwardly toward the second end 68 of the airfoil 52. All the values in Table I are given at room temperature and are non-filleted.

By defining X and Y coordinate values at selected locations in a Z direction normal to the X, Y plane, the profile section or airfoil shape of the airfoil 52 of the turbine nozzle 50, at each Z distance along the length of the airfoil 52 can be ascertained. By connecting the X and Y values with smooth continuing arcs, each profile section at each distance Z is fixed. The airfoil profiles of the various surface locations between the distances Z are determined by smoothly connecting the adjacent profile sections to one another to form the airfoil profile.

The Table I values are generated and shown to four decimal places for determining the profile of the airfoil. As the turbine nozzle surface heats up during operation of the turbine 14, mechanical stresses and elevated temperatures will cause a change in the X, Y and Z values. Accordingly, the values for the profile given in Table I represent ambient, non-operating or non-hot conditions (e.g., room temperature) and are for an uncoated airfoil.

There are typical manufacturing tolerances as well as coatings which must be accounted for in the actual profile of the airfoil 52. Each section is joined smoothly with the other sections to form the complete airfoil shape. It will therefore be appreciated that +/− typical manufacturing tolerances, i.e., +/− values, including any coating thicknesses, are additive to the X and Y values given in Table I below. Accordingly, a distance of +/−5% in a direction normal to any surface location along the airfoil profile defines an airfoil profile envelope for this particular turbine nozzle airfoil design and turbine, i.e., a range of variation between measured points on the actual airfoil surface at nominal cold or room temperature and the ideal position of those points as given in the Table below at the same temperature. The data is scalable and the geometry pertains to all aerodynamic scales, at above and/or below 3000 RPM. The design of the airfoil 52 of the turbine nozzle 50 is robust to this range of variation without impairment of mechanical and aerodynamic functions.

TABLE I SUCTION-SIDE PRESSURE-SIDE X (%) Y (%) Z (%) N X (%) Y (%) Z (%) 0.0000 0.0000 0.0000 1 0.0000 0.0000 0.0000 −0.1248 −0.8295 0.0000 2 0.2295 0.6248 0.0000 −0.0814 −1.6674 0.0000 3 0.5705 1.1961 0.0000 0.1016 −2.4868 0.0000 4 1.0062 1.7000 0.0000 0.3938 −3.2736 0.0000 5 1.5147 2.1302 0.0000 0.7721 −4.0240 0.0000 6 2.0767 2.4884 0.0000 1.2178 −4.7357 0.0000 7 2.6752 2.7814 0.0000 1.7186 −5.4101 0.0000 8 3.2992 3.0155 0.0000 2.2643 −6.0496 0.0000 9 3.9403 3.1977 0.0000 2.8473 −6.6543 0.0000 10 4.5922 3.3341 0.0000 3.4620 −7.2271 0.0000 11 5.2519 3.4302 0.0000 4.1039 −7.7682 0.0000 12 5.9163 3.4884 0.0000 4.7705 −8.2806 0.0000 13 6.5822 3.5140 0.0000 5.4574 −8.7636 0.0000 14 7.2488 3.5109 0.0000 6.1643 −9.2186 0.0000 15 7.9147 3.4868 0.0000 6.8868 −9.6465 0.0000 16 8.5814 3.4574 0.0000 7.6256 −10.0473 0.0000 17 9.2473 3.4357 0.0000 8.3791 −10.4194 0.0000 18 9.9140 3.4225 0.0000 9.1450 −10.7636 0.0000 19 10.5806 3.4163 0.0000 9.9240 −11.0783 0.0000 20 11.2473 3.4163 0.0000 10.7155 −11.3620 0.0000 21 11.9140 3.4225 0.0000 11.5163 −11.6147 0.0000 22 12.5806 3.4364 0.0000 12.3271 −11.8349 0.0000 23 13.2465 3.4574 0.0000 13.1465 −12.0225 0.0000 24 13.9124 3.4860 0.0000 13.9721 −12.1775 0.0000 25 14.5783 3.5217 0.0000 14.8031 −12.3000 0.0000 26 15.2434 3.5651 0.0000 15.6388 −12.3884 0.0000 27 15.9085 3.6163 0.0000 16.4775 −12.4442 0.0000 28 16.5721 3.6752 0.0000 17.3171 −12.4667 0.0000 29 17.2357 3.7419 0.0000 18.1574 −12.4566 0.0000 30 17.8984 3.8163 0.0000 18.9961 −12.4140 0.0000 31 18.5597 3.8992 0.0000 19.8333 −12.3395 0.0000 32 19.2202 3.9907 0.0000 20.6667 −12.2341 0.0000 33 19.8791 4.0899 0.0000 21.4961 −12.0984 0.0000 34 20.5372 4.1977 0.0000 22.3194 −11.9326 0.0000 35 21.1938 4.3140 0.0000 23.1372 −11.7380 0.0000 36 21.8481 4.4388 0.0000 23.9473 −11.5155 0.0000 37 22.5016 4.5721 0.0000 24.7496 −11.2667 0.0000 38 23.1527 4.7147 0.0000 25.5434 −10.9907 0.0000 39 23.8023 4.8659 0.0000 26.3279 −10.6899 0.0000 40 24.4496 5.0256 0.0000 27.1031 −10.3659 0.0000 41 25.0938 5.1946 0.0000 27.8682 −10.0178 0.0000 42 25.7364 5.3729 0.0000 28.6225 −9.6481 0.0000 43 26.3760 5.5605 0.0000 29.3659 −9.2566 0.0000 44 27.0132 5.7566 0.0000 30.0984 −8.8450 0.0000 45 27.6473 5.9620 0.0000 30.8194 −8.4140 0.0000 46 28.2783 6.1767 0.0000 31.5295 −7.9643 0.0000 47 28.9062 6.4008 0.0000 32.2279 −7.4969 0.0000 48 29.5310 6.6341 0.0000 32.9140 −7.0124 0.0000 49 30.1519 6.8767 0.0000 33.5884 −6.5109 0.0000 50 30.7690 7.1287 0.0000 34.2512 −5.9946 0.0000 51 31.3829 7.3899 0.0000 34.9023 −5.4636 0.0000 52 31.9922 7.6605 0.0000 35.5419 −4.9186 0.0000 53 32.5969 7.9403 0.0000 36.1698 −4.3597 0.0000 54 33.1977 8.2287 0.0000 36.7853 −3.7884 0.0000 55 33.7946 8.5271 0.0000 37.3899 −3.2047 0.0000 56 34.3860 8.8341 0.0000 37.9822 −2.6093 0.0000 57 34.9729 9.1504 0.0000 38.5643 −2.0023 0.0000 58 35.5543 9.4760 0.0000 39.1341 −1.3853 0.0000 59 36.1318 9.8101 0.0000 39.6938 −0.7581 0.0000 60 36.7031 10.1527 0.0000 40.2419 −0.1217 0.0000 61 37.2698 10.5047 0.0000 40.7798 0.5240 0.0000 62 37.8310 10.8643 0.0000 41.3070 1.1783 0.0000 63 38.3860 11.2326 0.0000 41.8240 1.8403 0.0000 64 38.9364 11.6093 0.0000 42.3310 2.5101 0.0000 65 39.4806 11.9946 0.0000 42.8287 3.1876 0.0000 66 40.0194 12.3876 0.0000 43.3178 3.8713 0.0000 67 40.5519 12.7884 0.0000 43.7969 4.5605 0.0000 68 41.0783 13.1969 0.0000 44.2682 5.2566 0.0000 69 41.5992 13.6132 0.0000 44.7318 5.9574 0.0000 70 42.1140 14.0372 0.0000 45.1868 6.6636 0.0000 71 42.6225 14.4682 0.0000 45.6349 7.3744 0.0000 72 43.1248 14.9062 0.0000 46.0752 8.0899 0.0000 73 43.6209 15.3519 0.0000 46.5093 8.8101 0.0000 74 44.1109 15.8039 0.0000 46.9357 9.5333 0.0000 75 44.5946 16.2620 0.0000 47.3558 10.2612 0.0000 76 45.0721 16.7271 0.0000 47.7698 10.9922 0.0000 77 45.5442 17.1984 0.0000 48.1775 11.7271 0.0000 78 46.0093 17.6760 0.0000 48.5798 12.4651 0.0000 79 46.4682 18.1597 0.0000 48.9767 13.2054 0.0000 80 46.9209 18.6488 0.0000 49.3682 13.9488 0.0000 81 47.3674 19.1442 0.0000 49.7543 14.6953 0.0000 82 47.8078 19.6442 0.0000 50.1364 15.4434 0.0000 83 48.2419 20.1504 0.0000 50.5132 16.1946 0.0000 84 48.6705 20.6605 0.0000 50.8860 16.9481 0.0000 85 49.0930 21.1767 0.0000 51.2543 17.7031 0.0000 86 49.5093 21.6969 0.0000 51.6186 18.4605 0.0000 87 49.9194 22.2225 0.0000 51.9791 19.2194 0.0000 88 50.3233 22.7535 0.0000 52.3349 19.9806 0.0000 89 50.7209 23.2884 0.0000 52.6876 20.7434 0.0000 90 51.1124 23.8279 0.0000 53.0364 21.5078 0.0000 91 51.4977 24.3721 0.0000 53.3822 22.2736 0.0000 92 51.8775 24.9194 0.0000 53.7248 23.0403 0.0000 93 52.2527 25.4713 0.0000 54.0643 23.8093 0.0000 94 52.6217 26.0256 0.0000 54.4008 24.5791 0.0000 95 52.9868 26.5837 0.0000 54.7349 25.3496 0.0000 96 53.3450 27.1465 0.0000 55.0659 26.1225 0.0000 97 53.6969 27.7124 0.0000 55.3946 26.8961 0.0000 98 54.0550 28.2736 0.0000 55.6341 27.6930 0.0000 99 54.6295 28.5806 0.0000 55.2581 28.4124 0.0000 100 55.2581 28.4124 0.0000 −0.6806 −0.5589 10.0000 1 −0.6806 −0.5589 10.0000 −0.8512 −1.4039 10.0000 2 −0.4109 0.0628 10.0000 −0.8388 −2.2651 10.0000 3 −0.0318 0.6240 10.0000 −0.6729 −3.1116 10.0000 4 0.4372 1.1132 10.0000 −0.3860 −3.9256 10.0000 5 0.9729 1.5279 10.0000 −0.0070 −4.7008 10.0000 6 1.5566 1.8729 10.0000 0.4457 −5.4357 10.0000 7 2.1744 2.1535 10.0000 0.9558 −6.1326 10.0000 8 2.8140 2.3783 10.0000 1.5132 −6.7915 10.0000 9 3.4698 2.5527 10.0000 2.1101 −7.4147 10.0000 10 4.1349 2.6822 10.0000 2.7403 −8.0054 10.0000 11 4.8078 2.7729 10.0000 3.3984 −8.5643 10.0000 12 5.4837 2.8271 10.0000 4.0806 −9.0930 10.0000 13 6.1612 2.8496 10.0000 4.7853 −9.5922 10.0000 14 6.8395 2.8419 10.0000 5.5085 −10.0628 10.0000 15 7.5178 2.8124 10.0000 6.2496 −10.5062 10.0000 16 8.1953 2.7806 10.0000 7.0054 −10.9233 10.0000 17 8.8736 2.7574 10.0000 7.7767 −11.3116 10.0000 18 9.5512 2.7426 10.0000 8.5620 −11.6713 10.0000 19 10.2302 2.7341 10.0000 9.3597 −12.0008 10.0000 20 10.9085 2.7326 10.0000 10.1698 −12.3000 10.0000 21 11.5868 2.7380 10.0000 10.9907 −12.5667 10.0000 22 12.2651 2.7519 10.0000 11.8217 −12.8008 10.0000 23 12.9426 2.7729 10.0000 12.6612 −13.0016 10.0000 24 13.6209 2.8016 10.0000 13.5085 −13.1682 10.0000 25 14.2984 2.8388 10.0000 14.3620 −13.3000 10.0000 26 14.9752 2.8837 10.0000 15.2194 −13.3969 10.0000 27 15.6512 2.9364 10.0000 16.0806 −13.4589 10.0000 28 16.3271 2.9977 10.0000 16.9434 −13.4860 10.0000 29 17.0016 3.0674 10.0000 17.8070 −13.4783 10.0000 30 17.6760 3.1457 10.0000 18.6690 −13.4357 10.0000 31 18.3481 3.2333 10.0000 19.5295 −13.3597 10.0000 32 19.0202 3.3295 10.0000 20.3860 −13.2496 10.0000 33 19.6899 3.4341 10.0000 21.2372 −13.1078 10.0000 34 20.3589 3.5481 10.0000 22.0829 −12.9333 10.0000 35 21.0264 3.6713 10.0000 22.9217 −12.7287 10.0000 36 21.6915 3.8039 10.0000 23.7519 −12.4938 10.0000 37 22.3550 3.9457 10.0000 24.5744 −12.2295 10.0000 38 23.0163 4.0969 10.0000 25.3868 −11.9380 10.0000 39 23.6752 4.2581 10.0000 26.1899 −11.6202 10.0000 40 24.3318 4.4279 10.0000 26.9814 −11.2760 10.0000 41 24.9860 4.6085 10.0000 27.7628 −10.9078 10.0000 42 25.6372 4.7977 10.0000 28.5318 −10.5155 10.0000 43 26.2853 4.9977 10.0000 29.2891 −10.1016 10.0000 44 26.9310 5.2070 10.0000 30.0349 −9.6659 10.0000 45 27.5729 5.4264 10.0000 30.7682 −9.2109 10.0000 46 28.2109 5.6558 10.0000 31.4891 −8.7357 10.0000 47 28.8457 5.8946 10.0000 32.1977 −8.2419 10.0000 48 29.4775 6.1434 10.0000 32.8930 −7.7302 10.0000 49 30.1047 6.4023 10.0000 33.5767 −7.2023 10.0000 50 30.7271 6.6705 10.0000 34.2473 −6.6589 10.0000 51 31.3457 6.9488 10.0000 34.9054 −6.1000 10.0000 52 31.9605 7.2372 10.0000 35.5512 −5.5264 10.0000 53 32.5698 7.5341 10.0000 36.1845 −4.9395 10.0000 54 33.1744 7.8419 10.0000 36.8054 −4.3403 10.0000 55 33.7744 8.1581 10.0000 37.4140 −3.7279 10.0000 56 34.3698 8.4845 10.0000 38.0109 −3.1047 10.0000 57 34.9589 8.8202 10.0000 38.5961 −2.4698 10.0000 58 35.5434 9.1651 10.0000 39.1698 −1.8240 10.0000 59 36.1217 9.5194 10.0000 39.7326 −1.1690 10.0000 60 36.6946 9.8822 10.0000 40.2829 −0.5039 10.0000 61 37.2628 10.2543 10.0000 40.8233 0.1698 10.0000 62 37.8240 10.6341 10.0000 41.3527 0.8512 10.0000 63 38.3806 11.0233 10.0000 41.8721 1.5419 10.0000 64 38.9302 11.4202 10.0000 42.3806 2.2388 10.0000 65 39.4744 11.8248 10.0000 42.8798 2.9434 10.0000 66 40.0124 12.2380 10.0000 43.3698 3.6543 10.0000 67 40.5450 12.6589 10.0000 43.8512 4.3713 10.0000 68 41.0705 13.0868 10.0000 44.3233 5.0938 10.0000 69 41.5907 13.5233 10.0000 44.7876 5.8217 10.0000 70 42.1039 13.9667 10.0000 45.2442 6.5550 10.0000 71 42.6116 14.4171 10.0000 45.6930 7.2922 10.0000 72 43.1124 14.8744 10.0000 46.1341 8.0349 10.0000 73 43.6078 15.3380 10.0000 46.5690 8.7806 10.0000 74 44.0961 15.8085 10.0000 46.9969 9.5302 10.0000 75 44.5791 16.2853 10.0000 47.4178 10.2845 10.0000 76 45.0550 16.7682 10.0000 47.8333 11.0411 10.0000 77 45.5256 17.2574 10.0000 48.2426 11.8016 10.0000 78 45.9891 17.7519 10.0000 48.6457 12.5651 10.0000 79 46.4473 18.2527 10.0000 49.0434 13.3310 10.0000 80 46.8992 18.7589 10.0000 49.4364 14.1000 10.0000 81 47.3450 19.2705 10.0000 49.8248 14.8713 10.0000 82 47.7845 19.7868 10.0000 50.2078 15.6450 10.0000 83 48.2186 20.3085 10.0000 50.5868 16.4209 10.0000 84 48.6465 20.8341 10.0000 50.9612 17.1992 10.0000 85 49.0690 21.3651 10.0000 51.3310 17.9791 10.0000 86 49.4860 21.9008 10.0000 51.6977 18.7612 10.0000 87 49.8961 22.4403 10.0000 52.0597 19.5450 10.0000 88 50.3008 22.9853 10.0000 52.4178 20.3302 10.0000 89 50.7000 23.5341 10.0000 52.7729 21.1171 10.0000 90 51.0930 24.0868 10.0000 53.1240 21.9062 10.0000 91 51.4798 24.6442 10.0000 53.4729 22.6961 10.0000 92 51.8620 25.2047 10.0000 53.8178 23.4876 10.0000 93 52.2388 25.7690 10.0000 54.1605 24.2798 10.0000 94 52.6109 26.3357 10.0000 54.5000 25.0736 10.0000 95 52.9783 26.9062 10.0000 54.8372 25.8690 10.0000 96 53.3403 27.4798 10.0000 55.1713 26.6651 10.0000 97 53.6961 28.0581 10.0000 55.5023 27.4620 10.0000 98 54.0465 28.6380 10.0000 55.7116 28.2868 10.0000 99 54.5775 29.0357 10.0000 55.2364 28.9690 10.0000 100 55.2364 28.9690 10.0000 −1.3884 −1.2457 20.0000 1 −1.3884 −1.2457 20.0000 −1.5535 −2.1147 20.0000 2 −1.1147 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36.8465 9.1853 50.0000 40.6209 −2.1550 50.0000 60 37.4450 9.6341 50.0000 41.1977 −1.3915 50.0000 61 38.0364 10.0922 50.0000 41.7612 −0.6186 50.0000 62 38.6209 10.5589 50.0000 42.3140 0.1620 50.0000 63 39.1992 11.0341 50.0000 42.8550 0.9512 50.0000 64 39.7698 11.5171 50.0000 43.3853 1.7473 50.0000 65 40.3341 12.0085 50.0000 43.9047 2.5512 50.0000 66 40.8915 12.5070 50.0000 44.4140 3.3605 50.0000 67 41.4419 13.0132 50.0000 44.9132 4.1767 50.0000 68 41.9860 13.5271 50.0000 45.4031 4.9984 50.0000 69 42.5233 14.0473 50.0000 45.8845 5.8248 50.0000 70 43.0535 14.5752 50.0000 46.3566 6.6566 50.0000 71 43.5775 15.1085 50.0000 46.8209 7.4938 50.0000 72 44.0946 15.6488 50.0000 47.2775 8.3341 50.0000 73 44.6054 16.1953 50.0000 47.7256 9.1791 50.0000 74 45.1093 16.7481 50.0000 48.1674 10.0279 50.0000 75 45.6078 17.3062 50.0000 48.6016 10.8806 50.0000 76 46.0992 17.8705 50.0000 49.0295 11.7357 50.0000 77 46.5837 18.4395 50.0000 49.4512 12.5946 50.0000 78 47.0628 19.0140 50.0000 49.8659 13.4566 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80.0000 15 3.7930 −1.0574 80.0000 2.9054 −16.1023 80.0000 16 4.5946 −1.0124 80.0000 3.8442 −16.5031 80.0000 17 5.3961 −0.9574 80.0000 4.7953 −16.8736 80.0000 18 6.1969 −0.8953 80.0000 5.7581 −17.2124 80.0000 19 6.9977 −0.8279 80.0000 6.7318 −17.5186 80.0000 20 7.7977 −0.7535 80.0000 7.7155 −17.7915 80.0000 21 8.5969 −0.6713 80.0000 8.7085 −18.0295 80.0000 22 9.3953 −0.5822 80.0000 9.7085 −18.2318 80.0000 23 10.1922 −0.4853 80.0000 10.7163 −18.3969 80.0000 24 10.9891 −0.3798 80.0000 11.7287 −18.5233 80.0000 25 11.7845 −0.2667 80.0000 12.7457 −18.6124 80.0000 26 12.5783 −0.1457 80.0000 13.7651 −18.6620 80.0000 27 13.3713 −0.0155 80.0000 14.7860 −18.6721 80.0000 28 14.1620 0.1240 80.0000 15.8062 −18.6419 80.0000 29 14.9519 0.2721 80.0000 16.8248 −18.5721 80.0000 30 15.7395 0.4295 80.0000 17.8395 −18.4636 80.0000 31 16.5256 0.5961 80.0000 18.8496 −18.3147 80.0000 32 17.3093 0.7729 80.0000 19.8527 −18.1271 80.0000 33 18.0907 0.9589 80.0000 20.8481 −17.9016 80.0000 34 18.8698 1.1550 80.0000 21.8341 −17.6395 80.0000 35 19.6457 1.3620 80.0000 22.8109 −17.3403 80.0000 36 20.4194 1.5783 80.0000 23.7752 −17.0070 80.0000 37 21.1899 1.8062 80.0000 24.7271 −16.6388 80.0000 38 21.9574 2.0442 80.0000 25.6659 −16.2380 80.0000 39 22.7209 2.2922 80.0000 26.5907 −15.8062 80.0000 40 23.4814 2.5519 80.0000 27.5008 −15.3442 80.0000 41 24.2380 2.8225 80.0000 28.3961 −14.8535 80.0000 42 24.9907 3.1039 80.0000 29.2752 −14.3349 80.0000 43 25.7388 3.3961 80.0000 30.1388 −13.7907 80.0000 44 26.4822 3.6992 80.0000 30.9860 −13.2217 80.0000 45 27.2217 4.0140 80.0000 31.8178 −12.6295 80.0000 46 27.9558 4.3403 80.0000 32.6326 −12.0147 80.0000 47 28.6853 4.6775 80.0000 33.4310 −11.3791 80.0000 48 29.4093 5.0256 80.0000 34.2132 −10.7233 80.0000 49 30.1271 5.3845 80.0000 34.9791 −10.0488 80.0000 50 30.8403 5.7550 80.0000 35.7295 −9.3566 80.0000 51 31.5473 6.1364 80.0000 36.4636 −8.6465 80.0000 52 32.2481 6.5287 80.0000 37.1814 −7.9217 80.0000 53 32.9434 6.9318 80.0000 37.8845 −7.1814 80.0000 54 33.6318 7.3457 80.0000 38.5713 −6.4264 80.0000 55 34.3140 7.7705 80.0000 39.2434 −5.6581 80.0000 56 34.9891 8.2054 80.0000 39.9000 −4.8767 80.0000 57 35.6574 8.6504 80.0000 40.5426 −4.0829 80.0000 58 36.3194 9.1062 80.0000 41.1705 −3.2783 80.0000 59 36.9736 9.5721 80.0000 41.7837 −2.4628 80.0000 60 37.6217 10.0473 80.0000 42.3837 −1.6364 80.0000 61 38.2620 10.5326 80.0000 42.9698 −0.8008 80.0000 62 38.8946 11.0271 80.0000 43.5426 0.0442 80.0000 63 39.5202 11.5310 80.0000 44.1023 0.8977 80.0000 64 40.1388 12.0442 80.0000 44.6504 1.7589 80.0000 65 40.7496 12.5659 80.0000 45.1868 2.6271 80.0000 66 41.3527 13.0961 80.0000 45.7116 3.5023 80.0000 67 41.9481 13.6357 80.0000 46.2264 4.3845 80.0000 68 42.5364 14.1829 80.0000 46.7295 5.2721 80.0000 69 43.1163 14.7380 80.0000 47.2233 6.1659 80.0000 70 43.6891 15.3016 80.0000 47.7078 7.0643 80.0000 71 44.2550 15.8721 80.0000 48.1822 7.9682 80.0000 72 44.8124 16.4504 80.0000 48.6488 8.8760 80.0000 73 45.3628 17.0357 80.0000 49.1070 9.7884 80.0000 74 45.9054 17.6279 80.0000 49.5566 10.7047 80.0000 75 46.4411 18.2271 80.0000 49.9992 11.6240 80.0000 76 46.9690 18.8326 80.0000 50.4349 12.5473 80.0000 77 47.4891 19.4442 80.0000 50.8636 13.4744 80.0000 78 48.0023 20.0620 80.0000 51.2853 14.4039 80.0000 79 48.5085 20.6860 80.0000 51.7016 15.3357 80.0000 80 49.0078 21.3155 80.0000 52.1109 16.2705 80.0000 81 49.5000 21.9504 80.0000 52.5155 17.2078 80.0000 82 49.9845 22.5907 80.0000 52.9140 18.1473 80.0000 83 50.4628 23.2364 80.0000 53.3078 19.0891 80.0000 84 50.9341 23.8868 80.0000 53.6969 20.0333 80.0000 85 51.3992 24.5419 80.0000 54.0806 20.9791 80.0000 86 51.8574 25.2023 80.0000 54.4605 21.9264 80.0000 87 52.3093 25.8659 80.0000 54.8357 22.8760 80.0000 88 52.7543 26.5349 80.0000 55.2070 23.8264 80.0000 89 53.1938 27.2078 80.0000 55.5744 24.7791 80.0000 90 53.6271 27.8837 80.0000 55.9388 25.7326 80.0000 91 54.0543 28.5643 80.0000 56.2984 26.6876 80.0000 92 54.4752 29.2481 80.0000 56.6558 27.6442 80.0000 93 54.8907 29.9357 80.0000 57.0093 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−1.5047 90.0000 0.4868 −15.9442 90.0000 14 2.1860 −1.4132 90.0000 1.4326 −16.3845 90.0000 15 3.0039 −1.3326 90.0000 2.3930 −16.7915 90.0000 16 3.8217 −1.2473 90.0000 3.3659 −17.1682 90.0000 17 4.6395 −1.1543 90.0000 4.3504 −17.5147 90.0000 18 5.4558 −1.0566 90.0000 5.3450 −17.8295 90.0000 19 6.2721 −0.9550 90.0000 6.3488 −18.1124 90.0000 20 7.0876 −0.8488 90.0000 7.3612 −18.3628 90.0000 21 7.9023 −0.7364 90.0000 8.3822 −18.5791 90.0000 22 8.7155 −0.6178 90.0000 9.4093 −18.7612 90.0000 23 9.5287 −0.4938 90.0000 10.4426 −18.9070 90.0000 24 10.3403 −0.3628 90.0000 11.4798 −19.0155 90.0000 25 11.1512 −0.2248 90.0000 12.5209 −19.0876 90.0000 26 11.9612 −0.0806 90.0000 13.5636 −19.1217 90.0000 27 12.7690 0.0705 90.0000 14.6062 −19.1178 90.0000 28 13.5760 0.2295 90.0000 15.6488 −19.0752 90.0000 29 14.3814 0.3953 90.0000 16.6891 −18.9938 90.0000 30 15.1853 0.5698 90.0000 17.7248 −18.8744 90.0000 31 15.9868 0.7527 90.0000 18.7566 −18.7171 90.0000 32 16.7868 0.9442 90.0000 19.7814 −18.5209 90.0000 33 17.5845 1.1434 90.0000 20.7977 −18.2884 90.0000 34 18.3806 1.3519 90.0000 21.8054 −18.0186 90.0000 35 19.1736 1.5690 90.0000 22.8031 −17.7124 90.0000 36 19.9636 1.7953 90.0000 23.7891 −17.3721 90.0000 37 20.7519 2.0318 90.0000 24.7628 −16.9977 90.0000 38 21.5364 2.2775 90.0000 25.7233 −16.5907 90.0000 39 22.3178 2.5333 90.0000 26.6698 −16.1519 90.0000 40 23.0961 2.7984 90.0000 27.6016 −15.6829 90.0000 41 23.8713 3.0744 90.0000 28.5186 −15.1845 90.0000 42 24.6426 3.3605 90.0000 29.4194 −14.6581 90.0000 43 25.4093 3.6566 90.0000 30.3039 −14.1054 90.0000 44 26.1721 3.9636 90.0000 31.1721 −13.5279 90.0000 45 26.9310 4.2814 90.0000 32.0240 −12.9256 90.0000 46 27.6853 4.6093 90.0000 32.8589 −12.3000 90.0000 47 28.4341 4.9481 90.0000 33.6775 −11.6527 90.0000 48 29.1783 5.2984 90.0000 34.4791 −10.9853 90.0000 49 29.9178 5.6589 90.0000 35.2643 −10.2977 90.0000 50 30.6512 6.0302 90.0000 36.0326 −9.5922 90.0000 51 31.3791 6.4124 90.0000 36.7837 −8.8690 90.0000 52 32.1016 6.8054 90.0000 37.5194 −8.1287 90.0000 53 32.8178 7.2101 90.0000 38.2380 −7.3729 90.0000 54 33.5279 7.6248 90.0000 38.9411 −6.6016 90.0000 55 34.2318 8.0504 90.0000 39.6279 −5.8163 90.0000 56 34.9287 8.4868 90.0000 40.2992 −5.0178 90.0000 57 35.6194 8.9333 90.0000 40.9550 −4.2070 90.0000 58 36.3031 9.3907 90.0000 41.5953 −3.3829 90.0000 59 36.9791 9.8581 90.0000 42.2217 −2.5481 90.0000 60 37.6488 10.3357 90.0000 42.8326 −1.7031 90.0000 61 38.3109 10.8233 90.0000 43.4295 −0.8473 90.0000 62 38.9651 11.3209 90.0000 44.0124 0.0178 90.0000 63 39.6124 11.8287 90.0000 44.5814 0.8922 90.0000 64 40.2519 12.3457 90.0000 45.1388 1.7744 90.0000 65 40.8837 12.8721 90.0000 45.6829 2.6643 90.0000 66 41.5078 13.4078 90.0000 46.2155 3.5612 90.0000 67 42.1240 13.9527 90.0000 46.7364 4.4651 90.0000 68 42.7318 14.5062 90.0000 47.2473 5.3752 90.0000 69 43.3326 15.0682 90.0000 47.7465 6.2907 90.0000 70 43.9248 15.6388 90.0000 48.2364 7.2124 90.0000 71 44.5093 16.2171 90.0000 48.7163 8.1380 90.0000 72 45.0853 16.8039 90.0000 49.1876 9.0690 90.0000 73 45.6543 17.3984 90.0000 49.6504 10.0047 90.0000 74 46.2147 18.0000 90.0000 50.1039 10.9434 90.0000 75 46.7667 18.6093 90.0000 50.5504 11.8868 90.0000 76 47.3116 19.2256 90.0000 50.9891 12.8333 90.0000 77 47.8481 19.8488 90.0000 51.4209 13.7829 90.0000 78 48.3767 20.4783 90.0000 51.8457 14.7357 90.0000 79 48.8984 21.1147 90.0000 52.2643 15.6915 90.0000 80 49.4116 21.7566 90.0000 52.6775 16.6496 90.0000 81 49.9178 22.4054 90.0000 53.0837 17.6101 90.0000 82 50.4163 23.0589 90.0000 53.4845 18.5736 90.0000 83 50.9070 23.7186 90.0000 53.8806 19.5388 90.0000 84 51.3907 24.3837 90.0000 54.2713 20.5062 90.0000 85 51.8674 25.0543 90.0000 54.6574 21.4752 90.0000 86 52.3364 25.7295 90.0000 55.0380 22.4465 90.0000 87 52.7992 26.4093 90.0000 55.4155 23.4194 90.0000 88 53.2543 27.0946 90.0000 55.7876 24.3938 90.0000 89 53.7031 27.7837 90.0000 56.1566 25.3698 90.0000 90 54.1457 28.4767 90.0000 56.5217 26.3465 90.0000 91 54.5814 29.1736 90.0000 56.8837 27.3256 90.0000 92 55.0109 29.8752 90.0000 57.2419 28.3054 90.0000 93 55.4341 30.5806 90.0000 57.5961 29.2868 90.0000 94 55.8504 31.2899 90.0000 57.9473 30.2690 90.0000 95 56.2612 32.0023 90.0000 58.2953 31.2527 90.0000 96 56.6659 32.7178 90.0000 58.6411 32.2372 90.0000 97 57.0643 33.4372 90.0000 58.9837 33.2225 90.0000 98 57.4581 34.1597 90.0000 59.2527 34.2225 90.0000 99 57.8915 34.8527 90.0000 58.6612 35.0085 90.0000 100 58.6612 35.0085 90.0000 −8.0543 −7.0085 100.0000 1 −8.0543 −7.0085 100.0000 −8.2853 −8.0426 100.0000 2 −7.6946 −6.2426 100.0000 −8.2457 −9.1008 100.0000 3 −7.2116 −5.5481 100.0000 −7.9620 −10.1225 100.0000 4 −6.6364 −4.9271 100.0000 −7.4884 −11.0729 100.0000 5 −5.9930 −4.3767 100.0000 −6.8798 −11.9426 100.0000 6 −5.2992 −3.8915 100.0000 −6.1744 −12.7357 100.0000 7 −4.5682 −3.4636 100.0000 −5.3977 −13.4605 100.0000 8 −3.8093 −3.0876 100.0000 −4.5682 −14.1240 100.0000 9 −3.0287 −2.7589 100.0000 −3.6992 −14.7341 100.0000 10 −2.2318 −2.4736 100.0000 −2.7969 −15.2961 100.0000 11 −1.4209 −2.2271 100.0000 −1.8698 −15.8140 100.0000 12 −0.6008 −2.0163 100.0000 −0.9209 −16.2922 100.0000 13 0.2279 −1.8403 100.0000 0.0457 −16.7333 100.0000 14 1.0620 −1.6938 100.0000 1.0271 −17.1395 100.0000 15 1.8984 −1.5636 100.0000 2.0217 −17.5132 100.0000 16 2.7357 −1.4326 100.0000 3.0271 −17.8558 100.0000 17 3.5713 −1.2969 100.0000 4.0419 −18.1690 100.0000 18 4.4070 −1.1574 100.0000 5.0659 −18.4519 100.0000 19 5.2426 −1.0163 100.0000 6.0984 −18.7031 100.0000 20 6.0767 −0.8729 100.0000 7.1380 −18.9217 100.0000 21 6.9109 −0.7256 100.0000 8.1837 −19.1078 100.0000 22 7.7442 −0.5736 100.0000 9.2349 −19.2605 100.0000 23 8.5767 −0.4186 100.0000 10.2907 −19.3783 100.0000 24 9.4085 −0.2589 100.0000 11.3504 −19.4612 100.0000 25 10.2395 −0.0946 100.0000 12.4116 −19.5093 100.0000 26 11.0698 0.0752 100.0000 13.4736 −19.5209 100.0000 27 11.8984 0.2496 100.0000 14.5357 −19.4961 100.0000 28 12.7264 0.4295 100.0000 15.5961 −19.4341 100.0000 29 13.5527 0.6155 100.0000 16.6543 −19.3357 100.0000 30 14.3775 0.8070 100.0000 17.7078 −19.2016 100.0000 31 15.2008 1.0054 100.0000 18.7566 −19.0302 100.0000 32 16.0225 1.2109 100.0000 19.7984 −18.8225 100.0000 33 16.8426 1.4233 100.0000 20.8318 −18.5791 100.0000 34 17.6605 1.6426 100.0000 21.8574 −18.3000 100.0000 35 18.4767 1.8698 100.0000 22.8721 −17.9860 100.0000 36 19.2907 2.1047 100.0000 23.8760 −17.6388 100.0000 37 20.1016 2.3481 100.0000 24.8674 −17.2574 100.0000 38 20.9109 2.6000 100.0000 25.8465 −16.8442 100.0000 39 21.7163 2.8605 100.0000 26.8116 −16.4000 100.0000 40 22.5194 3.1295 100.0000 27.7612 −15.9248 100.0000 41 23.3194 3.4085 100.0000 28.6969 −15.4209 100.0000 42 24.1163 3.6961 100.0000 29.6163 −14.8891 100.0000 43 24.9093 3.9938 100.0000 30.5194 −14.3302 100.0000 44 25.6984 4.3016 100.0000 31.4070 −13.7450 100.0000 45 26.4829 4.6194 100.0000 32.2767 −13.1364 100.0000 46 27.2643 4.9473 100.0000 33.1302 −12.5031 100.0000 47 28.0411 5.2853 100.0000 33.9667 −11.8481 100.0000 48 28.8124 5.6341 100.0000 34.7860 −11.1721 100.0000 49 29.5798 5.9938 100.0000 35.5884 −10.4752 100.0000 50 30.3411 6.3636 100.0000 36.3729 −9.7597 100.0000 51 31.0977 6.7450 100.0000 37.1411 −9.0256 100.0000 52 31.8488 7.1372 100.0000 37.8922 −8.2736 100.0000 53 32.5938 7.5395 100.0000 38.6264 −7.5062 100.0000 54 33.3326 7.9535 100.0000 39.3434 −6.7225 100.0000 55 34.0651 8.3783 100.0000 40.0442 −5.9240 100.0000 56 34.7915 8.8147 100.0000 40.7295 −5.1116 100.0000 57 35.5116 9.2612 100.0000 41.3977 −4.2860 100.0000 58 36.2240 9.7194 100.0000 42.0504 −3.4481 100.0000 59 36.9295 10.1876 100.0000 42.6876 −2.5977 100.0000 60 37.6279 10.6667 100.0000 43.3093 −1.7364 100.0000 61 38.3186 11.1566 100.0000 43.9155 −0.8643 100.0000 62 39.0023 11.6574 100.0000 44.5085 0.0178 100.0000 63 39.6775 12.1682 100.0000 45.0868 0.9085 100.0000 64 40.3450 12.6899 100.0000 45.6519 1.8085 100.0000 65 41.0047 13.2209 100.0000 46.2039 2.7163 100.0000 66 41.6558 13.7628 100.0000 46.7442 3.6310 100.0000 67 42.2992 14.3140 100.0000 47.2721 4.5527 100.0000 68 42.9341 14.8744 100.0000 47.7884 5.4814 100.0000 69 43.5605 15.4450 100.0000 48.2938 6.4163 100.0000 70 44.1783 16.0240 100.0000 48.7884 7.3558 100.0000 71 44.7876 16.6124 100.0000 49.2736 8.3008 100.0000 72 45.3884 17.2093 100.0000 49.7496 9.2512 100.0000 73 45.9806 17.8147 100.0000 50.2155 10.2054 100.0000 74 46.5643 18.4287 100.0000 50.6736 11.1643 100.0000 75 47.1395 19.0504 100.0000 51.1240 12.1264 100.0000 76 47.7054 19.6806 100.0000 51.5659 13.0930 100.0000 77 48.2636 20.3178 100.0000 52.0008 14.0620 100.0000 78 48.8124 20.9628 100.0000 52.4279 15.0349 100.0000 79 49.3535 21.6147 100.0000 52.8488 16.0101 100.0000 80 49.8860 22.2729 100.0000 53.2636 16.9884 100.0000 81 50.4109 22.9380 100.0000 53.6729 17.9690 100.0000 82 50.9264 23.6101 100.0000 54.0752 18.9519 100.0000 83 51.4341 24.2876 100.0000 54.4729 19.9372 100.0000 84 51.9341 24.9713 100.0000 54.8651 20.9240 100.0000 85 52.4256 25.6612 100.0000 55.2519 21.9140 100.0000 86 52.9093 26.3566 100.0000 55.6349 22.9047 100.0000 87 53.3860 27.0574 100.0000 56.0124 23.8977 100.0000 88 53.8543 27.7628 100.0000 56.3860 24.8922 100.0000 89 54.3163 28.4729 100.0000 56.7566 25.8884 100.0000 90 54.7705 29.1876 100.0000 57.1225 26.8853 100.0000 91 55.2178 29.9070 100.0000 57.4845 27.8845 100.0000 92 55.6581 30.6302 100.0000 57.8434 28.8845 100.0000 93 56.0915 31.3581 100.0000 58.1984 29.8853 100.0000 94 56.5171 32.0907 100.0000 58.5496 30.8884 100.0000 95 56.9364 32.8264 100.0000 58.8984 31.8915 100.0000 96 57.3488 33.5659 100.0000 59.2442 32.8961 100.0000 97 57.7550 34.3093 100.0000 59.5868 33.9016 100.0000 98 58.1558 35.0558 100.0000 59.9008 34.9147 100.0000 99 58.6256 35.7496 100.0000 59.4287 35.8000 100.0000 100 59.4287 35.8000 100.0000

It will also be appreciated that the airfoil 52 disclosed in the above Table I may be scaled up or down geometrically for use in other similar turbine designs. Consequently, the coordinate values set forth in Table I may be scaled upwardly or downwardly such that the airfoil profile shape remains unchanged. A scaled version of the coordinates in Table I would be represented by X, Y and Z coordinate values of Table I, with the X, Y and Z non-dimensional coordinate values converted to inches, multiplied or divided by a constant number.

An important term in this disclosure is “profile”. The profile is the range of the variation between measured points on an airfoil surface and the ideal position listed in Table I. The actual profile on a manufactured turbine nozzle will be different than those in Table I and the design is robust to this variation meaning that mechanical and aerodynamic function are not impaired. As noted above, a + or −5% profile tolerance is used herein. The X, Y and Z values are all non-dimensionalized relative to the airfoil height.

The disclosed airfoil shape optimizes and is specific to the machine conditions and specifications. It provides a unique profile to achieve 1) interaction between other stages in the turbine 10; 2) aerodynamic efficiency; and 3) normalized aerodynamic and mechanical nozzle or airfoil loadings. The disclosed loci of points defined in Table I allow the gas turbine 12 or any other suitable turbine to run in an efficient, safe and smooth manner. As also noted, any scale of the disclosed airfoil 52 may be adopted as long as 1) interaction between other stages in the pressure turbine 10; 2) aerodynamic efficiency; and 3) normalized aerodynamic and mechanical nozzle or airfoil loadings are maintained in the scaled turbine.

The airfoil 52 described herein thus improves overall gas turbine 12 efficiency. Specifically, the airfoil 52 provides a desired turbine efficiency lapse rate (ISO, hot, cold, part load, etc.). The airfoil 52 also meets all aeromechanics and stress requirements. The turbine nozzle 50 described herein has very specific aerodynamic design requirements such as an upstream bucket radial back pressure (i.e., work splits) and radial velocity triangles for the downstream turbine rotors 150, 250, 350 and 450. Significant cross-functional design effort was require to meet these design goals. The airfoil 52 of the turbine nozzle 50 thus is of a specific shape to meet aerodynamic, mechanical, and heat transfer requirements in an efficient and cost effective manner.

This written description uses examples to disclose the invention, including the best mode, and also to enable any person skilled in the art to practice the invention, including making and using any devices or systems and performing any incorporated methods. The patentable scope of the invention is defined by the claims, and may include other examples that occur to those skilled in the art. Such other examples are intended to be within the scope of the claims if they have structural elements that do not differ from the literal language of the claims, or if they include equivalent structural elements with insubstantial differences from the literal languages of the claims. 

What is claimed is:
 1. A turbine nozzle comprising: an airfoil shape, the airfoil shape having a nominal profile substantially in accordance with Cartesian coordinate values of X, Y and Z set forth in Table I, wherein the Cartesian coordinate values of X, Y and Z are non-dimensional values from 0% to 100% convertible to dimensional distances in inches by multiplying the Cartesian coordinate values of X, Y and Z by a height of the airfoil in inches, and wherein X and Y, when connected by smooth continuing arcs, define airfoil profile sections at each distance Z, the airfoil profile sections at Z distances being joined smoothly with one another to form a complete airfoil shape.
 2. The turbine nozzle of claim 1, wherein the turbine nozzle forms part of a stage of a turbine.
 3. The turbine nozzle of claim 1, wherein the turbine nozzle is a third stage turbine nozzle.
 4. The turbine nozzle of claim 1, wherein the airfoil shape lies in an envelope within at least one of +/−5% and/or +/−5% of a chord length in a direction normal to any airfoil surface location.
 5. The turbine nozzle of claim 1, wherein a height of the turbine nozzle is about 5 inches to about 20 inches (about 12 centimeters to about 51 centimeters) in length.
 6. The turbine nozzle of claim 1, wherein the X, Y and Z distances are scalable as a function of the same constant or number to provide a scaled-up or scaled-down airfoil.
 7. A turbine nozzle comprising: an airfoil having a suction-side uncoated nominal airfoil profile substantially in accordance with suction-side Cartesian coordinate values of X, Y and Z set forth in Table I, wherein the Cartesian coordinate values of X, Y and Z are non-dimensional values from 0% to 100% convertible to dimensional distances in inches by multiplying the Cartesian coordinate values of X, Y and Z by a height of the airfoil in inches, and wherein X and Y, when connected by smooth continuing arcs, define airfoil profile sections at each Z distance, the airfoil profile sections at the Z distances being joined smoothly with one another to form a complete suction-side airfoil shape, the X, Y and Z distances being scalable as a function of the same constant or number to provide a scaled-up or scaled-down airfoil.
 8. The turbine nozzle of claim 7, wherein the turbine nozzle forms part of a stage of a turbine.
 9. The turbine nozzle of claim 7, wherein the turbine nozzle is a third stage turbine nozzle.
 10. The turbine nozzle of claim 7, wherein the airfoil shape lies in an envelope within at least one of +/−5% and/or +/−5% of a chord length in a direction normal to any airfoil surface location.
 11. The turbine nozzle of claim 7, wherein a height of the turbine nozzle is about 5 inches to about 20 inches (about 12 centimeters to about 51 centimeters) in length.
 12. A turbine comprising: a plurality of turbine nozzles, each of the turbine nozzles comprising an airfoil having an airfoil shape, the airfoil having a nominal profile substantially in accordance with Cartesian coordinate values of X, Y and Z set forth in Table I, wherein the Cartesian coordinate values of X, Y and Z are non-dimensional values from 0% to 100% convertible to dimensional distances in inches by multiplying the Cartesian coordinate values of X, Y and Z by a height of the airfoil in inches, and wherein X and Y, when connected by smooth continuing arcs, define airfoil profile sections at each Z distance, the airfoil profile sections at Z distances being joined smoothly with one another to form a complete airfoil shape.
 13. The turbine of claim 12, wherein the plurality of turbine nozzles forms part of a stage of a turbine.
 14. The turbine of claim 12, wherein the plurality of turbine nozzles comprises a plurality of third stage turbine nozzles.
 15. The turbine of claim 12, wherein the airfoil shape lies in an envelope within at least one of +/−5% and/or +/−5% of a chord length in a direction normal to any airfoil surface location.
 16. The turbine of claim 12, wherein a height the turbine nozzle is about 5 inches to about 20 inches (about 12 centimeters to about 51 centimeters) in length.
 17. The turbine of claim 12, wherein the X, Y and Z distances are scalable as a function of the same constant or number to provide a scaled-up or scaled-down airfoil.
 18. The turbine of claim 12, wherein X represents a distance parallel to a turbine axis of rotation. 